For a certain amount of work,it takes john 6 hours less than tommy,if they both work together,it takes them 13 hours 20 minutes,how long will it take tommy alone to complete the work
T is tommy's time, so tommy's rate is 1/T
John's rate is 1/(t-6)
Ratecombined=rateTommy+rateJohn
job=rate*time
1=(1/t+ 1/(t+6)) 13 1/3
t(t+6)=(t+6 + t)40/3
t^2+66-80t/3-240/3=0
3t^2-806t-62=0 check that
t=(806+-sqrt(806^2+12*62))/6
solve, ignore the negative solution.
check my calcs, I did them in my head (no paper handy)
1/x + 1/(x+6) = 1/(40/3) = 3/40
times 40x(x+6)
40(x+6) + 40x = 3x(x+6)
3x^2 - 62x - 240 = 0
t = (62 ± √6724)/6
= 24 or -3/1/3 , but x > 0
so x = 24
tom alone can do it in 24 hrs, John alone can do it in 30 hours
check:
at combined time,
time of job = 1/(1/24+1/3))
= 1/(3/40) = 40/3
= 13 1/3 hrs
To determine how long it will take Tommy alone to complete the work, we can break down the given information into equations and solve them step by step.
Let's denote the time it takes Tommy to complete the work alone as T hours. Since John takes 6 hours less than Tommy, John would take (T - 6) hours to complete the work alone.
Now, we know that when they work together, it takes them 13 hours and 20 minutes, which is equivalent to (13 + 20/60) hours or (800/60) hours, respectively.
When they work together, their rates of work add up. So, the equation we can set up is:
1/T + 1/(T - 6) = 1/(13 + 20/60)
To solve this equation, we need to find a common denominator for all fractions. In this case, the common denominator is T(T - 6)(13 + 20/60), which we can simplify to T(T - 6)(13 + 1/3).
Multiplying both sides of the equation by the common denominator, we get:
(T - 6)(13 + 1/3) + T(13 + 1/3) = T(T - 6)
Now, we can simplify and solve for T:
13(T - 6) + (1/3)(T - 6) + 13T + (1/3)T = T^2 - 6T
Simplifying further:
13T - 78 + (1/3)T - 2 + 13T + (1/3)T = T^2 - 6T
Combine like terms:
(27 + 1/3)T - 80 = T^2 - 6T
Multiply through by 3 to eliminate the fraction:
81T + 1T - 240 = 3T^2 - 18T
Rearrange and simplify:
3T^2 - 103T + 240 = 0
Factor the equation or use the quadratic formula to find the solutions for T. In this case, the equation factors easily:
(T - 15)(3T - 16) = 0
Therefore, T = 15 or T = 16/3. Since time cannot be negative, the only valid solution is T = 15.
So, it will take Tommy alone 15 hours to complete the work.